Download Publication: γ-Regular-open sets and γ

Mathematical Theory and Modeling
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-Regular-Open Sets and -Extremally Disconnected Spaces
Baravan A. Asaad* Nazihah Ahmad and Zurni Omar
School of Quantitative Sciences, College of Arts and Sciences, University Utara Malaysia, 06010 Sintok
* E-mail of the corresponding author: [email protected]
Abstract
The aim of this paper is to introduce a new class of sets called γ-regular-open sets in topological spaces (X, τ)
with an operation γ on τ together with its complement which is γ-regular-closed. Also to define a new space
called γ-extremally disconnected, and to obtain several characterizations of γ-extremally disconnected spaces by
utilizing γ-regular-open sets and γ-regular-closed sets. Further γ-locally indiscrete and γ-hyperconnected spaces
have been defined.
Keywords: γ-regular-open set, γ-regular-closed set, γ-extremally disconnected space, γ-locally indiscrete space
and γ-hyperconnected space
1. Introduction
In 1979, Kasahara defined the concept of an operation on topological spaces and introduced α-closed graphs of
an operation. Ogata (1991) called the operation α as γ operation on τ and defined and investigated the concept of
operation-open sets, that is, γ-open sets. He defined the complement of a γ-open subset of a space X as γ-closed.
In addition, he also proved that the union of any collection of γ-open sets in a topological space (X, τ) is γ-open,
but the intersection of any two γ-open sets in a space X need not be a γ-open set. Further study by Krishnan and
Balachandran (2006a; 2006b) defined two types of sets called γ-preopen and γ-semiopen sets of a topological
space (X, τ), respectively. Kalaivani and Krishnan (2009) defined the notion of α-γ-open sets. Basu, Afsan and
Ghosh (2009) used the operation γ to introduce γ-β-open sets. Finally, Carpintero, Rajesh and Rosas (2012a)
defined the notion of γ-b-open sets of a topological space (X, τ).
2. Preliminaries
Throughout this paper, (X, τ) (or simply X) will always denote topological spaces on which no separation axioms
are assumed unless explicitly stated. A subset R of X is said to be regular open and regular closed if R =
Int(Cl(R)) and R = Cl(Int(R))), respectively (Steen & Seebach, 1978), where Int(R) and Cl(R) denotes the interior
of R and the closure of R, respectively.
Now we recall some definitions and results which will be used in the sequel.
Definition 2.1 (Ogata, 1991): Let (X, τ) be a topological space. An operation γ on the topology τ on X is a
mapping γ : τ → P(X) such that U ⊂ γ(U) for each U ∈ τ, where P(X) is the power set of X and γ(U) denotes the
value of γ at U.
Definition 2.2 (Ogata, 1991): A nonempty set R of X is said to be γ-open if for each x ∈ R, there exists an open
set U such that x ∈ U and γ(U) ⊂ R. The complement of a γ-open set is called a γ-closed.
Definition 2.3: Let R be any subset of a topological space (X, τ) and γ be an operation on τ. Then
1) the τγ-closure of R is defined as intersection of all γ-closed sets containing R. That is,
τγ-Cl(R) = ∩ {F : R ⊂ F, X \ F ∈ τγ } (Ogata, 1991).
2) the τγ-interior of R is defined as union of all γ-open sets contained in R. That is,
τγ-Int(R) = ∪ {U : U is a γ-open set and U ⊂ R} (Krishnan & Balachandran, 2006b).
Remark 2.4: For any subset R of a topological space (X, τ). Then:
1) R is γ-open if and only if τγ-Int(R) = R (Krishnan & Balachandran, 2006b).
2) R is γ-closed if and only if τγ-Cl(R) = R (Ogata, 1991).
Definition 2.5: Let (X, τ) be a topological space and γ be an operation on τ. A subset R of X is said to be:
1) α-γ-open if R ⊂ τγ-Int(τγ-Cl(τγ-Int(R))) (Kalaivani & Krishnan, 2009).
2) γ-preopen if R ⊂ τγ-Int(τγ-Cl(R)) (Krishnan & Balachandran, 2006a).
3) γ-semiopen if R ⊂ τγ-Cl(τγ-Int(R)) (Krishnan & Balachandran, 2006b).
4) γ-b-open if R ⊂ τγ-Cl(τγ-Int(R)) ∪ τγ-Int(τγ-Cl(R)) (Carpintero et al, 2012a).
5) γ-β-open if R ⊂ τγ-Cl(τγ-Int(τγ-Cl(R))) (Basu et al, 2009).
6) γ-clopen if it is both γ-open and γ-closed.
Definition 2.6: The complement of α-γ-open, γ-preopen, γ-semiopen, γ-b-open and γ-β-open set is said to be α-γclosed (Kalaivani & Krishnan, 2009), γ-preclosed (Krishnan & Balachandran, 2006a), γ-semiclosed (Krishnan &
Balachandran, 2006b), γ-b-closed (Carpintero et al, 2012a) and γ-β-closed (Basu et al, 2009), respectively.
The intersection of all γ-semiclosed sets of X containing a subset R of X is called the γ-semi-closure of R
and it is denoted by τγ-sCl(R).
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The family of all γ-open, α-γ-open, γ-preopen, γ-semiopen, γ-b-open, γ-β-open, γ-semiclosed, γ-bclosed, γ-β-closed and regular open subsets of a topological space (X, τ) is denoted by τγ, τα-γ, τγ-PO(X), τγSO(X), τγ-BO(X), τγ-βO(X), τγ-SC(X), τγ-BC(X), τγ-βC(X) and RO(X), respectively.
Definition 2.7 (Ogata, 1991): Let (X, τ) be any topological space. An operation γ is said to be regular if for every
open neighborhood U and V of each x ∈ X, there exists an open neighborhood W of x such that γ(W) ⊂ γ(U) ∩
γ(V).
Definition 2.8 (Ogata, 1991): A topological space (X, τ) with an operation γ on τ is said to be γ-regular if for
each x ∈ X and for each open neighborhood V of x, there exists an open neighborhood U of x such that γ(U) ⊂ V.
Remark 2.9: If a topological space (X, τ) is γ-regular, then τγ = τ (Ogata, 1991) and hence τγ-Int(R) = Int(R)
(Krishnan, 2003)
Theorem 2.10 (Krishnan & Balachandran, 2006b): Let R be any subset of a topological space (X, τ) and γ be a
regular operation on τ. Then τγ-sCl(R) = R ∪ τγ-Int(τγ-Cl(R)).
Lemma 2.11 (Krishnan & Balachandran, 2006a): For any subset R of a topological space (X, τ) and γ be an
operation on τ. The following statements are true.
1) τγ-Int(X\R) = X\τγ-Cl(R) and τγ-Cl(X\R) = X\τγ-Int(R).
2) τγ-Cl(R) = X\τγ-Int(X\R) and τγ-Int(R) = X\τγ-Cl(X\R).
Lemma 2.12: Let R and S be any subsets of a topological space (X, τ) and γ be an operation on τ. If R ∩ S = φ,
then τγ-Int(R) ∩ τγ-Cl(S) = φ and τγ-Cl(R) ∩ τγ-Int(S) = φ.
Proof: Obvious.
Lemma 2.13 (Krishnan & Balachandran, 2006a): Let (X, τ) be a topological space and γ be a regular operation
on τ. Then for every γ-open set U and every subset R of X, we have τγ-Cl(R) ∩ U ⊆ τγ-Cl(R ∩ U).
Definition 2.14 (Carpintero et al, 2012b): A subset D of a topological space (X, τ) is said to be:
1) γ-dense if τγ-Cl(D) = X.
2) γ-semi-dense if τγ-sCl(D) = X.
3. γ-Regular-Open Sets
In this section, we introduce a new class of sets called γ-regular-open sets. This class of sets lies strictly between
the classes of γ-clopen and γ-open sets.
Definition 3.1: A subset R of a topological space (X, τ) with an operation γ on τ is said to be γ-regular-open if R
= τγ-Int(τγ-Cl(R)). The complement of a γ-regular-open set is γ-regular-closed. Or equivalently, a subset R of a
space X is said to be γ-regular-closed if R = τγ-Cl(τγ-Int(R)). The class of all γ-regular-open and γ-regular-closed
subsets of a topological space (X, τ) is denoted by τγ-RO(X, τ) or τγ-RO(X) and τγ-RC(X, τ) or τγ-RC(X),
respectively.
Remark 3.2: It is clear from the definition that every γ-regular-open set is γ-open and every γ-clopen set is both
γ-regular-open and γ-regular-closed.
Converses of the above remark are not true. It can be seen from the following example.
Example 3.3: Let X = {a, b, c} and τ = {φ, X, {a}, {b}, {a, b}}. Define an operation γ : τ → P(X) by: γ(R) = R
for every R ∈ τ. Then τγ = τ and τγ-RO(X, τ) = {φ, X, {a}, {b}}. Then the set {a, b} is γ-open, but it is not γregular-open. Also the sets {a} and {a, c} are γ-regular-open and γ-regular-closed, respectively, but they are not
γ-clopen.
Remark 3.4: Let R be any subset of a topological space (X, τ) and γ be an operation on τ. Then:
1) R is γ-clopen if and only if it is γ-regular-open and γ-regular-closed.
2) R is γ-regular-open if and only if it is γ-preopen and γ-semiclosed.
3) R is γ-regular-closed if and only if it is γ-semiopen and γ-preclosed.
Proof: Follows from their definitions.
Theorem 3.5: The concept of γ-regular-open set and regular open set are independent. It is showing by the
following example.
Example 3.6: Let X = {a, b, c} and τ = {φ, X, {a}, {b}, {a, b}, {b, c}}. Let γ : τ → P(X) be an operation defined
as follows: For every R ∈ τ. Then
γ(R) =
{
R,
if R = {b}
R ∪ {a}, if R ≠ {b}
}
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Then τγ = {φ, X, {a}, {b}, {a, b}}, τγ-RO(X, τ) = {φ, X, {a}, {b}} and RO(X, τ) = {φ, X, {a}, {b, c}}. Then the
set {b} is γ-regular-open, but it is not regular open. Also the set {b, c} is regular open, but it is not γ-regularopen.
Theorem 3.7: If (X, τ) be a γ-regular space, then the concept of γ-regular-open set and regular open set coincide.
Proof: Follows from Remark 2.9.
Remark 3.8: The union and intersection of any two γ-regular-open (respectively, γ-regular-closed) sets need not
be γ-regular-open (respectively, γ-regular-closed) set. It is shown by the following example.
Example 3.9: In Example 3.6, the sets {a} and {b} (respectively, {a, c} and {b, c}) are γ-regular-open
(respectively, γ-regular-closed) sets, but {a} ∪ {b} = {a, b} (respectively, {a, c} ∩ {b, c} = {c}) is not γ-regularopen (respectively, γ-regular-closed) set.
Theorem 3.10: Let (X, τ) be a topological space and γ be a regular operation on τ. Then:
1) The intersection of two γ-regular-open sets is γ-regular-open.
2) The union of two γ-regular-closed sets is γ-regular-closed.
Proof: Straightforward from Corollary 3.28 and Corollary 3.29 and the fact that every γ-regular-open set is γopen and every γ-regular-closed set is γ-closed.
Theorem 3.11: For any subset R of a topological space (X, τ), then R ∈ τγ-βO(X) if and only if τγ-Cl(R) = τγCl(τγ-Int(τγ-Cl(R))).
Proof: Let R ∈ τγ-βO(X), then R ⊂ τγ-Cl(τγ-Int(τγ-Cl(R))). Then R ⊂ τγ-Cl(τγ-Int(τγ-Cl(R))) implies that τγ-Cl(R)
⊂ τγ-Cl(τγ-Int(τγ-Cl(R))) ⊂ τγ-Cl(R). Therefore, τγ-Cl(R) = τγ-Cl(τγ-Int(τγ-Cl(R))).
Conversely, suppose that τγ-Cl(R) = τγ-Cl(τγ-Int(τγ-Cl(R))). This implies that τγ-Cl(R) ⊂ τγ-Cl(τγ-Int(τγ-Cl(R))).
Then R ⊂ τγ-Cl(R) ⊂ τγ-Cl(τγ-Int(τγ-Cl(R))). Hence R ⊂ τγ-Cl(τγ-Int(τγ-Cl(R))). Therefore, R ∈ τγ-βO(X).
Theorem 3.12: For any subset R of a topological space (X, τ), then R ∈ τγ-SO(X) if and only if τγ-Cl(R) = τγCl(τγ-Int(R)).
Proof: The proof is similar to Theorem 3.11.
Proposition 3.13: For any subset R of a topological space (X, τ). If R ∈ τγ-SO(X), then τγ-Cl(R) ∈ τγ-RC(X).
Proof: The proof is immediate consequence of Theorem 3.12.
Corollary 3.14: For any subset R of a topological space (X, τ). Then
1) R ∈ τγ-SC(X) if and only if τγ-Int(R) = τγ-Int(τγ-Cl(R)).
2) if R ∈ τγ-SC(X), then τγ-Int(R) ∈ τγ-RO(X).
Theorem 3.15: Let R be any subset of a topological space (X, τ) and γ be an operation on τ. Then
1) R ∈ τγ-βO(X) if and only if τγ-Cl(R) ∈ τγ-RC(X).
2) R ∈ τγ-βO(X) if and only if τγ-Cl(R) ∈ τγ-SO(X).
3) R ∈ τγ-βO(X) if and only if τγ-Cl(R) ∈ τγ-βO(X).
4) R ∈ τγ-βO(X) if and only if τγ-Cl(R) ∈ τγ-BO(X).
Proof:
1) The proof is immediate consequence of Theorem 3.11.
2) Let R ∈ τγ-βO(X), then by (1), τγ-Cl(R) ∈ τγ-RC(X) ⊂ τγ-SO(X). So τγ-Cl(R) ∈ τγ-SO(X). On the other hand, let
τγ-Cl(R) ∈ τγ-SO(X). Then by Theorem 3.12, τγ-Cl(τγ-Cl(R)) = τγ-Cl(τγ-Int(τγ-Cl(τγ-Cl(R)))) which implies that
τγ-Cl(R) = τγ-Cl(τγ-Int(τγ-Cl(R))) and hence by Theorem 3.11, R ∈ τγ-βO(X).
3) Let R ∈ τγ-βO(X), then by (2), τγ-Cl(R) ∈ τγ-SO(X) ⊂ τγ-βO(X). So τγ-Cl(R) ∈ τγ-βO(X). On the other hand, let
τγ-Cl(R) ∈ τγ-βO(X), by (2), τγ-Cl(τγ-Cl(R)) ∈ τγ-SO(X). Since τγ-Cl(τγ-Cl(R)) = τγ-Cl(R). Then τγ-Cl(R) ∈ τγSO(X) and hence by (2), R ∈ τγ-βO(X).
4) Let R ∈ τγ-βO(X), then by (2), τγ-Cl(R) ∈ τγ-SO(X) ⊂ τγ-BO(X). So τγ-Cl(R) ∈ τγ-BO(X). On the other hand,
let τγ-Cl(R) ∈ τγ-BO(X) ⊂ τγ-βO(X), then τγ-Cl(R) ∈ τγ-βO(X) and hence by (3), R ∈ τγ-βO(X).
From Theorem 3.11 and Theorem 3.15, we have the following corollary.
Corollary 3.16: For any subset R of a topological space (X, τ). Then
1) R ∈ τγ-βC(X) if and only if τγ-Int(R) = τγ-Int(τγ-Cl(τγ-Int(R))).
2) R ∈ τγ-βC(X) if and only if τγ-Int(R) ∈ τγ-RO(X).
3) R ∈ τγ-βC(X) if and only if τγ-Int(R) ∈ τγ-SC(X).
4) R ∈ τγ-βC(X) if and only if τγ-Int(R) ∈ τγ-βC(X).
5) R ∈ τγ-βC(X) if and only if τγ-Int(R) ∈ τγ-BC(X).
Corollary 3.17: For any subset R of a topological space (X, τ). Then
1) If R ∈ τγ-RO(X), then τγ-Cl(R) ∈ τγ-RC(X) and τγ-Int(R) ∈ τγ-RO(X).
2) If R ∈ τγ-RC(X), then τγ-Cl(R) ∈ τγ-RC(X) and τγ-Int(R) ∈ τγ-RO(X).
Lemma 3.18: For any subset R of a topological space (X, τ), then τγ-Cl(τγ-Int(τγ-Cl(τγ-Int(R)))) = τγ-Cl(τγInt(R)).
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Proof: Since τγ-Int(τγ-Cl(τγ-Int(R))) ⊂ τγ-Cl(τγ-Int(R)). Then τγ-Cl(τγ-Int(τγ-Cl(τγ-Int(R)))) ⊂ τγ-Cl(τγ-Int(R)). On
the other hand, since τγ-Int(R) ⊂ τγ-Cl(τγ-Int(R)) implies that τγ-Int(R) ⊂ τγ-Int(τγ-Cl(τγ-Int(R))) and hence τγCl(τγ-Int(R)) ⊂ τγ-Cl(τγ-Int(τγ-Cl(τγ-Int(R))). Therefore, τγ-Cl(τγ-Int(τγ-Cl(τγ-Int(R)))) = τγ-Cl(τγ-Int(R)).
Lemma 3.19: For any subset R of a topological space (X, τ), then τγ-Int(τγ-Cl(τγ-Int(τγ-Cl(R)))) = τγ-Int(τγCl(R)).
Proof: The proof is similar to Lemma 3.18.
It is obvious from Lemma 3.18 and Lemma 3.19 that τγ-Cl(τγ-Int(R)) and τγ-Int(τγ-Cl(R)) are γ-regularclosed and γ-regular-open sets, respectively.
Lemma 3.20: Let R be any subset of a topological space (X, τ) and γ be a regular operation on τ. Then R ∈ τγPO(X) if and only if τγ-sCl(R) = τγ-Int(τγ-Cl(R)).
Proof: Let R ∈ τγ-PO(X), then R ⊂ τγ-Int(τγ-Cl(R) implies that τγ-sCl(R) ⊂ τγ-sCl(τγ-Int(τγ-Cl(R))). Since τγInt(τγ-Cl(R)) ∈ τγ-RO(X). But τγ-RO(X) ⊂ τγ-SC(X) in general, then τγ-Int(τγ-Cl(R)) ∈ τγ-SC(X) and hence τγsCl(τγ-Int(τγ-Cl(R))) = τγ-Int(τγ-Cl(R)). So τγ-sCl(R) ⊂ τγ-Int(τγ-Cl(R)). On the other hand, by Theorem 2.10, we
have τγ-sCl(R) = R ∪ τγ-Int(τγ-Cl(R)). Then τγ-Int(τγ-Cl(R)) ⊂ τγ-sCl(R). Therefore, τγ-sCl(R) = τγ-Int(τγ-Cl(R)).
Conversely, let τγ-sCl(R) = τγ-Int(τγ-Cl(R)), then τγ-sCl(R) ⊂ τγ-Int(τγ-Cl(R)) and hence R ⊂ τγ-sCl(R) ⊂ τγ-Int(τγCl(R)) which implies that R ⊂ τγ-Int(τγ-Cl(R)). Then R ∈τγ-PO(X).
Proposition 3.21: Let P and R be subsets of a topological space (X, τ) and γ be an operation on τ. Then P is γpreopen if and only if there exists a γ-regular open set R containing P such that τγ-Cl(P) = τγ-Cl(R).
Proof: Let P be a γ-preopen set, then P ⊂ τγ-Int(τγ-Cl(P)). Put R = τγ-Int(τγ-Cl(P)) is a γ-regular open set
containing P. Since P is γ-preopen, then P is γ-β-open. By Theorem 3.11, τγ-Cl(P) = τγ-Cl(τγ-Int(τγ-Cl(P))) = τγCl(R).
Conversely, suppose R be a γ-regular-open set and P be any subset such that P ⊂ R and τγ-Cl(P) = τγ-Cl(R). Then
P ⊂ τγ-Int(τγ-Cl(R)) and hence P ⊂ τγ-Int(τγ-Cl(P)). This means that P is γ-preopen set. This completes the proof.
Proposition 3.22: If S is both γ-semiopen and γ-semiclosed subset of a topological space (X, τ) with an operation
γ on τ and τγ-Cl(τγ-Int(S)) ⊂ τγ-Int(τγ-Cl(S)). Then S is both γ-regular-open and γ-regular-closed.
Proof: Clear.
Lemma 3.23: Let R be any subset of a topological space (X, τ) and γ be an operation on τ. Then the following
statements are equivalent:
1) R is γ-regular-open.
2) R is γ-open and γ-semiclosed.
3) R is α-γ-open and γ-semiclosed.
4) R is γ-preopen and γ-semiclosed.
5) R is γ-open and γ-β-closed.
6) R is α-γ-open and γ-β-closed.
Proof:
(1) ⇒ (2) Let R be γ-regular-open set. Since every γ-regular-open set is γ-open and every γ-regular open set is γsemiclosed. Then R is γ-open and γ-semiclosed.
(2) ⇒ (3) Let R be γ-open and γ-semiclosed set. Since every γ-open set is α-γ-open. Then R is α-γ-open and γsemiclosed.
(3) ⇒ (4) Let R be α-γ-open and γ-semiclosed set. Since every α-γ-open set is γ-preopen. Then A is γ-preopen
and γ-semiclosed.
(4) ⇒ (5) Let R be γ-preopen and γ-semiclosed set. Then R ⊂ τγ-Int(τγ-Cl(R)) and τγ-Int(τγ-Cl(R)) ⊂ R.
Therefore, we have R = τγ-Int(τγ-Cl(R)). Then R is γ-regular-open set and hence it is γ-open. Since every γsemiclosed set is γ-β-closed. Then R is γ-open and γ-β-closed.
(5) ⇒ (6) It is obvious since every γ-open set is α-γ-open.
(6) ⇒ (1) Let R be α-γ-open and γ-β-closed set. Then R ⊂ τγ-Int(τγ-Cl(τγ-Int(R))) and τγ-Int(τγ-Cl(τγ-Int(R))) ⊂ R.
Then τγ-Int(R) = τγ-Int(τγ-Cl(τγ-Int(R))) = R and hence τγ-Int(τγ-Cl(R)) = τγ-Int(τγ-Cl(τγ-Int(R))) = R. Therefore, R
is γ-regular-open set.
Corollary 3.24: Let S be any subset of a topological space (X, τ) and γ be an operation on τ. Then the following
statements are equivalent:
1) S is γ-regular-closed.
2) S is γ-closed and γ-semiopen.
3) S is α-γ-closed and γ-semiopen.
4) S is γ-preclosed and γ-semiopen.
5) S is γ-closed and γ-β-open.
6) S is α-γ-closed and γ-β-open.
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Proof: Similar to Lemma 3.23 taking R = X\S.
Lemma 3.25: Let R be any subset of a topological space (X, τ) and γ be an operation on τ. Then the following
statements are equivalent:
1) R is γ-clopen.
2) R is γ-regular-open and γ-regular-closed.
3) R is γ-open and α-γ-closed.
4) R is γ-open and γ-preclosed.
5) R is α-γ-open and γ-preclosed.
6) R is α-γ-open and γ-closed.
7) R is γ-preopen and γ-closed.
8) R is γ-preopen and α-γ-closed.
Proof:
(1) ⇔ (2) see Remark 3.4 (1).
The implications (2) ⇒ (3), (3) ⇒ (4), (4) ⇒ (5), (6) ⇒ (7) and (7) ⇒ (8) are obvious (see Figure 1).
(5) ⇒ (6) Let R be α-γ-open and γ-preclosed set. Then R ⊂ τγ-Int(τγ-Cl(τγ-Int(R))) and τγ-Cl(τγ-Int(R))) ⊂ R. This
implies that R = τγ-Int(τγ-Cl(τγ-Int(R))) and hence τγ-Cl(R) = τγ-Cl(τγ-Int(τγ-Cl(τγ-Int(R)))). By Lemma 3.18, we
get τγ-Cl(R) = τγ-Cl(τγ-Int(R)). Since τγ-Cl(τγ-Int(R))) ⊂ R, then τγ-Cl(R) ⊂ R. But in general R ⊂ τγ-Cl(R). Then
τγ-Cl(R) = R. It is obvious that R is γ-closed.
(8) ⇒ (1) Let R be γ-preopen and α-γ-closed. Then R ⊂ τγ-Int(τγ-Cl(R)) and τγ-Cl(τγ-Int(τγ-Cl(R))) ⊂ R.
Therefore, we have τγ-Cl(R) ⊂ τγ-Cl(τγ-Int(τγ-Cl(R))) ⊂ R and hence τγ-Cl(R) ⊂ R. But in general R ⊂ τγ-Cl(R).
Then τγ-Cl(R) = R. It is obvious that R is γ-closed. Since τγ-Cl(τγ-Int(τγ-Cl(R))) ⊂ R implies that τγ-Int(τγ-Cl(τγInt(τγ-Cl(R)))) ⊂ τγ-Int(R). Then by Lemma 3.19, R ⊂ τγ-Int(τγ-Cl(R)) ⊂ τγ-Int(R) and hence R ⊂ τγ-Int(R). But in
general τγ-Int(R) ⊂ R. Then τγ-Int(R) = R. It is obvious that R is γ-open. Therefore, R is γ-clopen.
Proposition 3.26: Let R and S be any two subsets of a topological space (X, τ) and γ be an operation on τ. Then:
τγ-Int(τγ-Cl(R ∩ S)) ⊂ τγ-Int(τγ-Cl(R)) ∩ τγ-Int(τγ-Cl(S))
Proof: The proof is obvious and hence it is omitted.
The converse of the above proposition is true when the operation γ is regular operation on τ and if one
of the set is γ-open in X, as shown by the following proposition.
Proposition 3.27: Let (X, τ) be a topological space and γ be a regular operation on τ. If R is γ-open subset of X
and S is any subset of X, then
τγ-Int(τγ-Cl(R)) ∩ τγ-Int(τγ-Cl(S)) = τγ-Int(τγ-Cl(R ∩ S))
Proof: It is enough to prove τγ-Int(τγ-Cl(R)) ∩ τγ-Int(τγ-Cl(S)) ⊂ τγ-Int(τγ-Cl(R ∩ S)) since the converse is
similar to Proposition 3.26. Since R is γ-open subset of a space X and γ is a regular operation on τ. Then by using
Lemma 2.13, we have
τγ-Int(τγ-Cl(R)) ∩ τγ-Int(τγ-Cl(S)) ⊂ τγ-Int[τγ-Cl(R) ∩ τγ-Int(τγ-Cl(S))]
⊂ τγ-Int(τγ-Cl[R ∩ τγ-Cl(S)]) ⊂ τγ-Int(τγ-Cl(R ∩ S))
So τγ-Int(τγ-Cl(R)) ∩ τγ-Int(τγ-Cl(S)) = τγ-Int(τγ-Cl(R ∩ S)). This completes the proof.
From Proposition 3.27, we have the following corollary.
Corollary 3.28: If R and S are γ-open subsets of a topological space (X, τ) and γ be a regular operation on τ, then
τγ-Int(τγ-Cl(R)) ∩ τγ-Int(τγ-Cl(S)) = τγ-Int(τγ-Cl(R ∩ S))
Corollary 3.29: If E and F are γ-closed subsets of a topological space (X, τ) and γ be a regular operation on τ,
then
τγ-Cl(τγ-Int(E)) ∪ τγ-Cl(τγ-Int(F)) = τγ-Cl(τγ-Int(E ∪ F))
Proof: the proof is similar to Corollary 3.28 taking R = X\E and S = X\F.
4. γ-Extremally Disconnected Spaces
In this section, we introduce a new space called γ-extremally disconnected, and to obtain several
characterizations of γ-extremally disconnected spaces by utilizing γ-regular-open sets and γ-regular-closed sets.
Definition 4.1: A topological space (X, τ) with an operation γ on τ is said to be γ-extremally disconnected if the
τγ-closure of every γ-open set of X is γ-open in X. Or equivalently, a space X is γ-extremally disconnected if the
τγ-interior of every γ-closed set of X is γ-closed in X.
In the following theorem, a space X is γ-extremally disconnected is equivalent to every two disjoint γopen sets of X have disjoint τγ-closures.
Theorem 4.2: A space X is γ-extremally disconnected if and only if τγ-Cl(R) ∩ τγ-Cl(S) = φ for every γ-open
subsets R and S of X with R ∩ S = φ.
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Proof: Suppose R and S are two γ-open subsets of a γ-extremally disconnected space X such that R ∩ S = φ.
Then by Lemma 2.12, τγ-Cl(R) ∩ S = φ which implies that τγ-Int(τγ-Cl(R)) ∩ τγ-Cl(S) = φ and hence τγ-Cl(R) ∩
τγ-Cl(S) = φ.
Conversely, let O be any γ-open subset of a space X, then X\O is γ-closed set and hence τγ-Int(X\O) is γ-open set
such that O ∩ τγ-Int(X\O) = φ. Then by hypothesis, we have τγ-Cl(O) ∩ τγ-Cl(τγ-Int(X\O)) = φ which implies that
τγ-Cl(O) ∩ τγ-Cl(X\τγ-Cl(O)) = φ and hence τγ-Cl(O) ∩ X\τγ-Int(τγ-Cl(O)) = φ. This means that τγ-Cl(O) ⊂ τγInt(τγ-Cl(O)). Since τγ-Int(τγ-Cl(O)) ⊂ τγ-Cl(O) in general. Then τγ-Cl(O) = τγ-Int(τγ-Cl(O)). So τγ-Cl(O) is γopen set in X. Therefore, X is γ-extremally disconnected space.
Theorem 4.3: Let (X, τ) be a topological space and γ be a regular operation on τ. Then the following are
equivalent:
1) X is γ-extremally disconnected.
2) τγ-Cl(R) ∩ τγ-Cl(S) = τγ-Cl(R ∩ S) for every γ-open subsets R and S of X.
3) τγ-Cl(R) ∩ τγ-Cl(S) = τγ-Cl(R ∩ S) for every γ-regular-open subsets R and S of X.
4) τγ-Int(E) ∪ τγ-Int(F) = τγ-Int(E ∪ F) for every γ-regular-closed subsets E and F of X.
5) τγ-Int(E) ∪ τγ-Int(F) = τγ-Int(E ∪ F) for every γ-closed subsets E and F of X.
Proof:
(1) ⇒ (2) Let R and S be any two γ-open subsets of a γ-extremally disconnected space X. Then by Corollary
3.28, τγ-Cl(R) ∩ τγ-Cl(S) = τγ-Int(τγ-Cl(R)) ∩ τγ-Int(τγ-Cl(S)) = τγ-Int(τγ-Cl(R ∩ S)) = τγ-Cl(R ∩ S).
(2) ⇒ (3) is clear since every γ-regular-open set is γ-open.
(3) ⇔ (4) Let E and F be two γ-regular-closed subsets of X. Then X\E and X\F are γ-regular-open sets. By (3)
and Lemma 2.11, we have
τγ-Cl(X\E) ∩ τγ-Cl(X\F) = τγ-Cl(X\E ∩ X\F)
⇔ X\τγ-Int(E) ∩ X\τγ-Int(F) = τγ-Cl(X\(E ∪ F))
⇔ X\(τγ-Int(E) ∪ τγ-Int(F)) = X\τγ-Int(E ∪ F)
⇔ τγ-Int(E) ∪ τγ-Int(F) = τγ-Int(E ∪ F).
(4) ⇒ (5) Let E and F be two γ-closed subsets of X. Then τγ-Cl(τγ-Int(E)) and τγ-Cl(τγ-Int(F)) are γ-regularclosed sets. Then by (4) and Corollary 3.29, we get
τγ-Int(τγ-Cl(τγ-Int(E))) ∪ τγ-Int(τγ-Cl(τγ-Int(F))) = τγ-Int[τγ-Cl(τγ-Int(E)) ∪ τγ-Cl(τγ-Int(F))] and hence τγ-Int(τγCl(τγ-Int(E))) ∪ τγ-Int(τγ-Cl(τγ-Int(F))) = τγ-Int(τγ-Cl(τγ-Int(E ∪ F))).
Since E and F are γ-closed subsets of X. Then by Lemma 3.19, we obtain
τγ-Int(τγ-Cl(E)) ∪ τγ-Int(τγ-Cl(F)) = τγ-Int(τγ-Cl(E ∪ F)). This implies that τγ-Int(E) ∪ τγ-Int(F) = τγ-Int(E ∪ F).
(5) ⇔ (2) the proof is similar to (3) ⇔ (4).
(2) ⇒ (1) let U be any γ-open subset of a space X, then X\U is γ-closed set and hence τγ-Int(X\U) is γ-open set.
Then by (2), we τγ-Cl(U) ∩ τγ-Cl(τγ-Int(X\U)) = τγ-Cl(U ∩ τγ-Int(X\U)) which implies that τγ-Cl(U) ∩ τγ-Cl(X\τγCl(U)) = τγ-Cl(φ) since U ∩ τγ-Int(X\U) = φ. Hence τγ-Cl(U) ∩ X\τγ-Int(τγ-Cl(U)) = φ. This means that τγ-Cl(U)
⊂ τγ-Int(τγ-Cl(U)). Since τγ-Int(τγ-Cl(U)) ⊂ τγ-Cl(U) in general. Then τγ-Cl(U) = τγ-Int(τγ-Cl(U)). So τγ-Cl(U) is
γ-open set in X. Therefore, a space X is γ-extremally disconnected.
Theorem 4.4: Let (X, τ) be a topological space and γ be a regular operation on τ. Then the following are
equivalent:
1) X is γ-extremally disconnected.
2) τγ-Cl(R) ∩ τγ-Cl(S) = φ for every γ-open subsets R and S of X with R ∩ S = φ.
3) τγ-Cl(R) ∩ τγ-Cl(S) = φ for every γ-regular-open subsets R and S of X with R ∩ S = φ.
Proof:
(1) ⇔ (2) see Theorem 4.3.
(2) ⇒ (3) since every γ-regular-open set is γ-open. Then the proof is clear.
(3) ⇒ (2) Let R and S be any two γ-open subsets of a space X such that R ∩ S = φ. Then by Lemma 2.12, R ∩ τγCl(S) = φ implies that τγ-Cl(R) ∩ τγ-Int(τγ-Cl(S)) = φ and hence τγ-Int(τγ-Cl(R)) ∩ τγ-Int(τγ-Cl(S)) = φ. Since τγInt(τγ-Cl(R)) and τγ-Int(τγ-Cl(S)) are two γ-regular-open sets. Then by (3), we obtain τγ-Cl(τγ-Int(τγ-Cl(R))) ∩ τγCl(τγ-Int(τγ-Cl(S))) = φ. Since τγ-Int(τγ-Cl(R)) and τγ-Int(τγ-Cl(S)) are two γ-regular-open sets, then τγ-Int(τγCl(R)) and τγ-Int(τγ-Cl(S)) are two γ-β-open sets. So by Theorem 3.11, we get τγ-Cl(R) ∩ τγ-Cl(S) = φ. This
completes the proof.
Theorem 4.5: A space X is γ-extremally disconnected if and only if τγ-Cl(R) ∩ τγ-Cl(τγ-Int(τγ-Cl(S))) = φ for
every γ-open subset R and every subset S of X with R ∩ S = φ.
Proof: see Theorem 4.4, since R and τγ-Int(τγ-Cl(S)) are two γ-open subsets of X such that R ∩ τγ-Int(τγ-Cl(S)) =
φ.
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Theorem 4.6: Let (X, τ) be a topological space and γ be a regular operation on τ. Then X is γ-extremally
disconnected if and only if τγ-Int(τγ-Cl(R)) ∪ τγ-Int(τγ-Cl(S)) = τγ-Int(τγ-Cl(R ∪ S)) for every γ-open subsets R
and S of X.
Proof: Let (X, τ) be a γ-extremally disconnected space and let R and S be any two γ-open subsets of X. Then τγCl(R) and τγ-Cl(S) are γ-closed subsets of X. So by Theorem 4.3 (5), we have τγ-Int(τγ-Cl(R)) ∪ τγ-Int(τγ-Cl(S)) =
τγ-Int(τγ-Cl(R) ∪ τγ-Cl(S)) = τγ-Int(τγ-Cl(R ∪ S)).
Conversely, let E and F be two γ-closed subsets of X. Then τγ-Int(E) and τγ-Int(F) are γ-open subsets of X. So by
hypothesis, τγ-Int(τγ-Cl(τγ-Int(E))) ∪ τγ-Int(τγ-Cl(τγ-Int(F))) = τγ-Int(τγ-Cl[τγ-Int(E) ∪ τγ-Int(F)]) = τγ-Int(τγCl(τγ-Int(E ∪ F))). Since E and F are γ-closed subsets of X. Then by Lemma 3.19, τγ-Int(τγ-Cl(E)) ∪ τγ-Int(τγCl(F)) = τγ-Int(τγ-Cl(E ∪ F)) and hence τγ-Int(E) ∪ τγ-Int(F) = τγ-Int(E ∪ F). Therefore, by Theorem 4.3 (5), X is
γ-extremally disconnected space.
Theorem 4.7: Let (X, τ) be a topological space and γ be a regular operation on τ. Then X is γ-extremally
disconnected if and only if τγ-Cl(τγ-Int(E)) ∩ τγ-Cl(τγ-Int(F)) = τγ-Cl(τγ-Int(E ∩ F)) for every γ-closed subsets E
and F of X.
Proof: Similar to Theorem 4.6 taking R = X\E and S = X\F.
Theorem 4.8: A space X is γ-extremally disconnected if and only if τγ-RO(X) = τγ-RC(X).
Proof: Obvious.
Theorem 4.9: Let (X, τ) be a topological space and γ be a regular operation on τ. Then the following are
equivalent:
1) X is γ-extremally disconnected.
2) R1 ∩ R2 is γ-regular-closed for all γ-regular-closed subsets R1 and R2 of X.
3) R1 ∪ R2 is γ-regular-open for all γ-regular-open subsets R1 and R2 of X.
Proof: The proof is directly from Theorem 3.10 and Theorem 4.8.
Theorem 4.10: The following statements are equivalent for any topological space (X, τ).
1) X is γ-extremally disconnected.
2) Every γ-regular-closed subset of X is γ-open in X.
3) Every γ-regular-closed subset of X is α-γ-open in X.
4) Every γ-regular-closed subset of X is γ-preopen in X.
5) Every γ-semiopen subset of X is α-γ-open in X.
6) Every γ-semiclosed subset of X is α-γ-closed in X.
7) Every γ-semiclosed subset of X is γ-preclosed in X.
8) Every γ-semiopen subset of X is γ-preopen in X.
9) Every γ-β-open subset of X is γ-preopen in X.
10) Every γ-β-closed subset of X is γ-preclosed in X.
11) Every γ-b-closed subset of X is γ-preclosed in X.
12) Every γ-b-open subset of X is γ-preopen in X.
13) Every γ-regular-open subset of X is γ-preclosed in X.
14) Every γ-regular-open subset of X is γ-closed in X.
15) Every γ-regular-open subset of X is α-γ-closed in X.
Proof:
(1) ⇒ (2) Let R be any γ-regular-closed subset of a γ-extremally disconnected space X. Then R = τγ-Cl(τγ-Int(R)).
Since R is γ-regular-closed set, then it is γ-closed and hence R = τγ-Cl(τγ-Int(R)) = τγ-Int(R). Therefore, R is γopen set in X.
The implications (2) ⇒ (3) and (3) ⇒ (4) are clear since every γ-open set is α-γ-open and every α-γ-open set is
γ-preopen.
(4) ⇒ (5) Let S be a γ-semiopen set. Then S ⊂ τγ-Cl(τγ-Int(S)). Since τγ-Cl(τγ-Int(S)) is γ-regular-closed set. Then
by (4), τγ-Cl(τγ-Int(S)) is γ-preopen and hence τγ-Cl(τγ-Int(S)) ⊂ τγ-Int(τγ-Cl(τγ-Cl(τγ-Int(S)))) = τγ-Int(τγ-Cl(τγInt(S))). So S ⊂ τγ-Int(τγ-Cl(τγ-Int(S))). Therefore, S is α-γ-open set.
The implications (5) ⇔ (6), (6) ⇒ (7), (7) ⇔ (8), (9) ⇔ (10), (10) ⇒ (11), (11) ⇔ (12) and (14) ⇒ (15) are
obvious.
(8) ⇒ (9) Let G be a γ-β-open set. Then by Theorem 3.15 (2), τγ-Cl(G) is γ-semiopen set. So by (8), τγ-Cl(G) is
γ-preopen set. So τγ-Cl(G) ⊂ τγ-Int(τγ-Cl(τγ-Cl(G))) = τγ-Int(τγ-Cl(G)) and hence G ⊂ τγ-Int(τγ-Cl(G)). Therefore,
G is γ-preopen set in X.
(12) ⇒ (13) Let H be a γ-regular-open set. Then H is γ-β-open set. By Theorem 3.15 (4), τγ-Cl(H) is γ-b-open
set. Then by (12), τγ-Cl(H) is γ-preopen. So τγ-Cl(H) ⊂ τγ-Int(τγ-Cl(τγ-Cl(H))) = τγ-Int(τγ-Cl(H)). Since H is γ-
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regular-open set. Then τγ-Cl(H) ⊂ H. Since H ⊂ τγ-Cl(H). Then τγ-Cl(H) = H. This means that H is γ-closed and
hence it is γ-preclosed.
(13) ⇒ (14) Let U be a γ-regular-open set. Then by (13), U is γ-preclosed set. So τγ-Cl(τγ-Int(U)) ⊂ U. Since U is
γ-regular-open set, then U is γ-open. Hence τγ-Cl(U) ⊂ U. But in general U ⊂ τγ-Cl(U). Then τγ-Cl(U) = U. This
means that U is γ-closed.
(15) ⇒ (1) Let V be any γ-open set of X. Then τγ-Int(τγ-Cl(V)) is γ-regular-open set. By (15), τγ-Int(τγ-Cl(V)) is
α-γ-closed. So τγ-Cl(τγ-Int(τγ-Cl(τγ-Int(τγ-Cl(V))))) ⊂ τγ-Int(τγ-Cl(V)). By Lemma 3.18, we get τγ-Cl(V) ⊂ τγInt(τγ-Cl(V)). But τγ-Int(τγ-Cl(V)) ⊂ τγ-Cl(V) in general. Then τγ-Cl(V) = τγ-Int(τγ-Cl(V)) and hence τγ-Cl(V) is γopen set of X. Therefore, X is γ-extremally disconnected space.
Theorem 4.11: The following conditions are equivalent for any topological space (X, τ).
1) X is γ-extremally disconnected.
2) The τγ-closure of every γ-β-open set of X is γ-regular-open in X.
3) The τγ-closure of every γ-b-open set of X is γ-regular-open in X.
4) The τγ-closure of every γ-semiopen set of X is γ-regular-open in X.
5) The τγ-closure of every α-γ-open set of X is γ-regular-open in X.
6) The τγ-closure of every γ-open set of X is γ-regular-open in X.
7) The τγ-closure of every γ-regular-open set of X is γ-regular-open in X.
8) The τγ-closure of every γ-preopen set of X is γ-regular-open in X.
Proof:
(1) ⇒ (2) Let R be a γ-β-open subset of a γ-extremally disconnected space X. Then by (1) and Lemma 3.11, we
have τγ-Cl(R) = τγ-Cl(τγ-Int(τγ-Cl(R))) = τγ-Int(τγ-Cl(R)) implies that τγ-Cl(R) = τγ-Int(τγ-Cl(R)) = τγ-Int(τγ-Cl(τγCl(R))). Hence τγ-Cl(R) in γ-regular-open set in X.
The implications (2) ⇒ (3), (3) ⇒ (4), (4) ⇒ (5), (5) ⇒ (6) and (6) ⇒ (7) are clear.
(7) ⇒ (8) Let P be any γ-preopen set of X, then τγ-Int(τγ-Cl(P)) is γ-regular-open. By (7), τγ-Cl(τγ-Int(τγ-Cl(P)))
is γ-regular-open set. So τγ-Cl(τγ-Int(τγ-Cl(P))) = τγ-Int(τγ-Cl(τγ-Cl(τγ-Int(τγ-Cl(P))))). Since every γ-preopen set
is γ-β-open. Then by Theorem 3.11 and Lemma 3.19, we have τγ-Cl(P) = τγ-Int(τγ-Cl(P)) = τγ-Int(τγ-Cl(τγCl(P))). Hence τγ-Cl(P) is γ-regular-open set in X.
(8) ⇒ (1) Let S be a γ-open set of X. Then S is γ-preopen and by (8), τγ-Cl(S) is γ-regular-open set in X. Then τγCl(S) is γ-open. Therefore, X is γ-extremally disconnected space.
Remark 4.12: γ-regular-closed set can be replaced by γ-regular-open set in Theorem 4.11 (this is because of
Theorem 4.8).
5. γ-Locally Indiscrete and γ-Hyperconnected Spaces
In this section, we introduce new types of spaces called γ-locally indiscrete and γ-hyperconnected. We give some
properties and characterizations of these spaces.
Definition 5.1: A topoplogical space (X, τ) with an operation γ on τ is said to be:
1) γ-locally indiscrete if every γ-open subset of X is γ-closed, or every γ-closed subset of X is γ-open.
2) γ-hyperconnected if every nonempty γ-open subset of X is γ-dense.
Theorem 5.2: Let (X, τ) be a topological space and γ be an operation on τ. Then the following are holds:
1) If X is γ-locally indiscrete, then X is γ-extremally disconnected.
2) If X is γ-hyperconnected, then X is γ-extremally disconnected.
Proof: Follows from their definitions.
Theorem 5.3: If (X, τ) is γ-locally indiscrete space, then the following statements are true:
1) Every γ-semiopen subset of X is γ-open and hence it is γ-closed.
2) Every γ-semiclosed subset of X is γ-closed and hence it is γ-open.
3) Every γ-open subset of X is γ-regular-open and hence it is γ-regular-closed.
4) Every γ-closed subset of X is γ-regular-closed and hence it is γ-regular-open.
5) Every γ-semiopen subset of X is γ-regular-open and hence it is γ-regular-closed.
6) Every γ-semiclosed subset of X is γ-regular-closed and hence it is γ-regular-open.
7) Every γ-β-open subset of X is γ-preopen.
8) Every γ-β-closed subset of X is γ-preclosed.
Proof:
1) Let S be any γ-semiopen subset of a γ-locally indiscrete space (X, τ), then S ⊂ τγ-Cl(τγ-Int(S)). Since τγ-Int(S)
is γ-open subset of X, then it is γ-closed. So τγ-Cl(τγ-Int(S)) = τγ-Int(S) implies that S ⊂ τγ-Int(S). But τγ-Int(S) ⊂
S. Then S = τγ-Int(S), this means that S is γ-open. Since a space X is γ-locally indiscrete, then S is γ-closed.
2) The proof is similar to part (1).
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3) Let O be any γ-open subset of a γ-locally indiscrete space (X, τ). Since every γ-open set is γ-closed, then τγCl(τγ-Int(O)) = O. This implies that O is a γ-regular-closed set.
4) The proof is similar to part (3).
5) Follows directly from (1) and (3).
6) Follows directly from (2) and (4).
7) Let P be a γ-β-open subset of a γ-locally indiscrete space (X, τ). Then P ⊂ τγ-Cl(τγ-Int(τγ-Cl(P))) ⊂ τγ-Int(τγCl(P)). Since τγ-Int(τγ-Cl(P)) is γ-open set and hence it is γ-closed in γ-locally indiscrete space X. Then τγ-Cl(τγInt(τγ-Cl(P))) = τγ-Int(τγ-Cl(P)). Then P ⊂ τγ-Int(τγ-Cl(P)). Hence P is γ-preopen set in X.
8) The proof is similar to part (7).
From Theorem 5.3, we have the following corollary.
Corollary 5.4: If (X, τ) is γ-locally indiscrete space, then
1) τγ-RO(X) = τγ = τα-γ = τγ-SO(X).
2) τγ-PO(X) = τγ-BO(X) = τγ-βO(X).
Theorem 5.5: A space (X, τ) is γ-hyperconnected if and only if τγ-RO(X) = {φ, X}.
Proof: In general φ and X are γ-regular-open subsets of a γ-hyperconnected space X. Let R be any nonempty
proper subset of X which is γ-regular open. Then R is γ-open set. Since X is γ-hyperconnected space. So τγ-Int(τγCl(R)) = τγ-Int(X) = X and hence R is γ-regular-open set in X. Contradiction. Therefore, τγ-RO(X) = {φ, X}.
Conversely, suppose that τγ-RO(X) = {φ, X} and let S be any nonempty γ-open subset of X. Then S is γ-β-open
set. By Theorem 3.11, τγ-Cl(S) = τγ-Cl(τγ-Int(τγ-Cl(S))). Since τγ-Int(τγ-Cl(S)) is γ-regular-open set and S is
nonempty γ-open set. Then τγ-Int(τγ-Cl(S)) should be X. Therefore, τγ-Cl(S) = τγ-Cl(τγ-Int(τγ-Cl(S))) = τγ-Cl(X) =
X. Then a space X is γ-hyperconnected.
Corollary 5.6: A space (X, τ) is γ-hyperconnected if and only if τγ-RC(X) = {φ, X}.
Proposition 5.7: If a space (X, τ) is γ-hyperconnected, then every nonempty γ-preopen subset of X is γ-semidense.
Proof: Let P be any nonempty γ-preopen subset of a γ-hyperconnected space X. By Lemma 3.19 and Lemma
3.20, we have τγ-sCl(P) = τγ-Int(τγ-Cl(P)) = τγ-Int(τγ-Cl(τγ-Int(τγ-Cl(P)))). Since τγ-Int(τγ-Cl(P)) is a nonempty γopen set and X is γ-hyperconnected space. Then τγ-Cl(τγ-Int(τγ-Cl(P))) = X and hence τγ-Int(τγ-Cl(τγ-Int(τγCl(P)))) = τγ-Int(X) = X. Thus τγ-sCl(P) = X. This completes the proof.
6. Conclusion
In this paper, we introduce a new class of sets called γ-regular-open sets in a topological space (X, τ) together
with its complement which is γ-regular-closed. Using these sets, we define γ-extremally disconnected space, and
to obtain several characterizations of γ-extremally disconnected spaces. Finally, γ-locally indiscrete and γhyperconnected spaces have been introduced.
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Berlin.
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By Remark 3.4, Definition 2.5 and Definition 2.6, we have the following figure.
Figure 1:
The relations between γ-regular-open set, γ-regular-closed set and various types of γ-sets.
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